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Properties

Label 56.48.1.gc.1
Level $56$
Index $48$
Genus $1$
Analytic rank $0$
Cusps $8$
$\Q$-cusps $0$

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Invariants

Level: $56$ $\SL_2$-level: $8$ Newform level: $64$
Index: $48$ $\PSL_2$-index:$48$
Genus: $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (none of which are rational) Cusp widths $4^{4}\cdot8^{4}$ Cusp orbits $4^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 8F1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 56.48.1.311

Level structure

$\GL_2(\Z/56\Z)$-generators: $\begin{bmatrix}1&54\\2&7\end{bmatrix}$, $\begin{bmatrix}9&1\\8&23\end{bmatrix}$, $\begin{bmatrix}15&17\\2&53\end{bmatrix}$, $\begin{bmatrix}45&3\\46&55\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 56.96.1-56.gc.1.1, 56.96.1-56.gc.1.2, 112.96.1-56.gc.1.1, 112.96.1-56.gc.1.2, 112.96.1-56.gc.1.3, 112.96.1-56.gc.1.4, 168.96.1-56.gc.1.1, 168.96.1-56.gc.1.2, 280.96.1-56.gc.1.1, 280.96.1-56.gc.1.2
Cyclic 56-isogeny field degree: $32$
Cyclic 56-torsion field degree: $768$
Full 56-torsion field degree: $64512$

Jacobian

Conductor: $2^{6}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 64.2.a.a

Models

Embedded model Embedded model in $\mathbb{P}^{3}$

$ 0 $ $=$ $ 3 y^{2} - 2 y z - 2 z^{2} - w^{2} $
$=$ $28 x^{2} - 2 y^{2} - y z - z^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} - 8 x^{2} y^{2} + 21 x^{2} z^{2} + 9 y^{4} - 84 y^{2} z^{2} + 196 z^{4} $
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Rational points

This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.

Maps between models of this curve

Birational map from embedded model to plane model:

$\displaystyle X$ $=$ $\displaystyle z$
$\displaystyle Y$ $=$ $\displaystyle 2x$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{7}w$

Maps to other modular curves

$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle -\frac{2^8}{3^4}\cdot\frac{1915325720yz^{11}+2052134700yz^{9}w^{2}+603669024yz^{7}w^{4}+11150244yz^{5}w^{6}-6794928yz^{3}w^{8}+510300yzw^{10}+1050958517z^{12}+1488041359z^{10}w^{2}+679775922z^{8}w^{4}+86500827z^{6}w^{6}-6839910z^{4}w^{8}-144585z^{2}w^{10}-30375w^{12}}{w^{4}(883568yz^{7}+568008yz^{5}w^{2}+109368yz^{3}w^{4}+6048yzw^{6}+487403z^{8}+480886z^{6}w^{2}+150675z^{4}w^{4}+16632z^{2}w^{6}+432w^{8})}$

Modular covers

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Cover information

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This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.24.1.ba.1 $8$ $2$ $2$ $1$ $0$ dimension zero
28.24.0.f.1 $28$ $2$ $2$ $0$ $0$ full Jacobian
56.24.0.cl.1 $56$ $2$ $2$ $0$ $0$ full Jacobian
56.24.0.dk.1 $56$ $2$ $2$ $0$ $0$ full Jacobian
56.24.0.ec.1 $56$ $2$ $2$ $0$ $0$ full Jacobian
56.24.1.y.1 $56$ $2$ $2$ $1$ $0$ dimension zero
56.24.1.bg.1 $56$ $2$ $2$ $1$ $0$ dimension zero

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus Rank Kernel decomposition
56.384.25.nu.1 $56$ $8$ $8$ $25$ $9$ $1^{20}\cdot2^{2}$
56.1008.73.bpi.1 $56$ $21$ $21$ $73$ $29$ $1^{16}\cdot2^{26}\cdot4$
56.1344.97.boo.1 $56$ $28$ $28$ $97$ $38$ $1^{36}\cdot2^{28}\cdot4$
112.96.3.kp.1 $112$ $2$ $2$ $3$ $?$ not computed
112.96.3.kr.1 $112$ $2$ $2$ $3$ $?$ not computed
112.96.3.nr.1 $112$ $2$ $2$ $3$ $?$ not computed
112.96.3.nt.1 $112$ $2$ $2$ $3$ $?$ not computed
168.144.9.fbe.1 $168$ $3$ $3$ $9$ $?$ not computed
168.192.9.bqr.1 $168$ $4$ $4$ $9$ $?$ not computed
280.240.17.bay.1 $280$ $5$ $5$ $17$ $?$ not computed
280.288.17.djo.1 $280$ $6$ $6$ $17$ $?$ not computed